def bruhat_graph(self, x, y):
"""
The Bruhat graph Gamma(x,y), defined if x <= y in the Bruhat order, has
as its vertices the Bruhat interval, {t | x <= t <= y}, and as its
edges the pairs u, v such that u = r.v where r is a reflection, that
is, a conjugate of a simple reflection.
Returns the Bruhat graph as a directed graph, with an edge u --> v
if and only if u < v in the Bruhat order, and u = r.v.
See:
Carrell, The Bruhat graph of a Coxeter group, a conjecture of Deodhar, and
rational smoothness of Schubert varieties. Algebraic groups and their
generalizations: classical methods (University Park, PA, 1991), 53--61,
Proc. Sympos. Pure Math., 56, Part 1, Amer. Math. Soc., Providence, RI, 1994.
EXAMPLES:
sage: W = WeylGroup("A3", prefix = "s")
sage: [s1,s2,s3] = W.simple_reflections()
sage: W.bruhat_graph(s1*s3,s1*s2*s3*s2*s1)
Digraph on 10 vertices
"""
g = self.bruhat_interval(x, y)
ref = self.reflections()
d = {}
for x in g:
d[x] = [y for y in g if x.length() < y.length() and ref.has_key(x*y.inverse())]
return DiGraph(d)
def graph(self):
"""
Convert ``self`` to a digraph
EXAMPLE::
sage: t1 = BinaryTree([[], None])
sage: t1.graph()
Digraph on 5 vertices
sage: t1 = BinaryTree([[], [[], None]])
sage: t1.graph()
Digraph on 9 vertices
sage: t1.graph().edges()
[(0, 1, None), (0, 4, None), (1, 2, None), (1, 3, None), (4, 5, None), (4, 8, None), (5, 6, None), (5, 7, None)]
"""
from sage.graphs.graph import DiGraph
res = DiGraph()
def rec(tr, idx):
if not tr:
return
else:
nbl = 2*tr[0].node_number() + 1
res.add_edges([[idx,idx+1], [idx,idx+1+nbl]])
rec(tr[0], idx + 1)
rec(tr[1], idx + nbl + 1)
rec(self, 0)
return res
def plot(self, edge_labels=True, graph_border=True, **kwds):
"""
Plot ``self``.
INPUT:
- ``edge_labels`` -- (default True) if edge label visible
- ``graph_border`` -- (default True) if graph border visible
OUTPUT:
Launched png viewer for Graphics object
EXAMPLES::
sage: from train_track.inverse_graph import GraphWithInverses
sage: G = GraphWithInverses([[0,0,'a'],[0,1,'b'],[1,0,'c']])
sage: G.plot()
Graphics object consisting of 11 graphics primitives
"""
return DiGraph.plot(DiGraph(self),
edge_labels=edge_labels,
graph_border=graph_border,
**kwds)
def cartan_matrix_as_graph(self, q=None):
from sage.graphs.graph import DiGraph
G = DiGraph()
G.add_vertices(self.idempotents())
for (e,f),coeff in self.cartan_matrix_as_table(q).iteritems():
G.add_edge(e,f, None if coeff == 1 else coeff)
return G
def quiver(self, edge_labels=True, index=False):
"""
Returns the quiver of ``self``
INPUT:
- ``edges_labels`` -- whether to use the quiver element
as label for the edges between the idempotents; if
False, this may lead to multiple edges when the monoid
is not generated by idempotents (default: True)
- ``index`` -- whether to index the vertices of the graph
by the indices of the simple modules rather than the
corresponding idempotents (default: False)
OUTPUT: the quiver of ``self``, as a graph with the
idempotents of this monoid as vertices
.. todo:: should index default to True?
.. todo:: use a meaningful example
EXAMPLES::
sage: import sage_semigroups.monoids.catalog as semigroups
sage: M = semigroups.NDPFMonoidPoset(Posets(3)[3])
sage: G = M.quiver()
sage: G
Digraph on 4 vertices
sage: G.edges()
[([1], [2], [3])]
sage: M.quiver(edge_labels=False).edges()
[([1], [2], None)]
sage: M.quiver(index=True).edges()
[(2, 1, [3])]
sage: M.quiver(index=True, edge_labels=False).edges()
[(2, 1, None)]
"""
from sage.graphs.graph import DiGraph
res = DiGraph()
if index:
res.add_vertices(self.simple_modules_index_set())
symbol = attrcall("symbol_index")
else:
res.add_vertices(self.idempotents())
symbol = attrcall("symbol")
for x in self.quiver_elements():
res.add_edge(symbol(x,"left"), symbol(x, "right"), x if edge_labels else None)
return res
def bruhat_graph(self, x, y):
r"""
Return the Bruhat graph as a directed graph, with an edge `u \to v`
if and only if `u < v` in the Bruhat order, and `u = r \cdot v`.
The Bruhat graph `\Gamma(x,y)`, defined if `x \leq y` in the
Bruhat order, has as its vertices the Bruhat interval
`\{ t | x \leq t \leq y \}`, and as its edges are the pairs
`(u, v)` such that `u = r \cdot v` where `r` is a reflection,
that is, a conjugate of a simple reflection.
REFERENCES:
Carrell, The Bruhat graph of a Coxeter group, a conjecture of Deodhar,
and rational smoothness of Schubert varieties. Algebraic groups and
their generalizations: classical methods (University Park, PA, 1991),
53--61, Proc. Sympos. Pure Math., 56, Part 1, Amer. Math. Soc.,
Providence, RI, 1994.
EXAMPLES::
sage: W = WeylGroup("A3", prefix="s")
sage: s1, s2, s3 = W.simple_reflections()
sage: G = W.bruhat_graph(s1*s3, s1*s2*s3*s2*s1); G
Digraph on 10 vertices
Check that the graph has the correct number of edges
(see :trac:`17744`)::
sage: len(G.edges())
16
"""
g = self.bruhat_interval(x, y)
ref = self.reflections()
d = {}
for u in g:
d[u] = [
v for v in g
if u.length() < v.length() and u * v.inverse() in ref
]
return DiGraph(d)
def subsets_lattice(self):
"""
Return the lattice of subsets ordered by containment.
EXAMPLES::
sage: X = Set([1,2,3])
sage: X.subsets_lattice()
Finite lattice containing 8 elements
sage: Y = Set()
sage: Y.subsets_lattice()
Finite lattice containing 1 elements
"""
if not self.is_finite():
raise NotImplementedError(
"this method is only implemented for finite sets")
from sage.combinat.posets.lattices import FiniteLatticePoset
from sage.graphs.graph import DiGraph
from sage.rings.integer import Integer
n = self.cardinality()
# list, contains at position 0 <= i < 2^n
# the i-th subset of self
subset_of_index = [
Set([self[i] for i in range(n) if v & (1 << i)])
for v in range(2**n)
]
# list, contains at position 0 <= i < 2^n
# the list of indices of all immediate supersets
upper_covers = [[
Integer(x | (1 << y)) for y in range(n) if not x & (1 << y)
] for x in range(2**n)]
# DiGraph, every subset points to all immediate supersets
D = DiGraph({
subset_of_index[v]: [subset_of_index[w] for w in upper_covers[v]]
for v in range(2**n)
})
# Lattice poset, defined by hasse diagram D
L = FiniteLatticePoset(hasse_diagram=D)
return L
def plot(self, edge_labels=True, graph_border=True, **kwds):
return DiGraph.plot(DiGraph(self),
edge_labels=edge_labels,
graph_border=graph_border,
**kwds)
def test_some_specific_examples():
r"""
Tests our ClassicalCrystalOfAlcovePaths against some specific examples.
EXAMPLES::
sage: sage.combinat.crystals.alcove_path.test_some_specific_examples() # long time (12s on sage.math, 2011)
G2 example passed.
C3 example passed.
B3 example 1 passed.
B3 example 2 passed.
True
"""
# This appears in Lenart.
C = ClassicalCrystalOfAlcovePaths(['G', 2], [0, 1])
G = C.digraph()
GT = DiGraph({
(): {
(0): 2
},
(0): {
(0, 8): 1
},
(0, 1): {
(0, 1, 7): 2
},
(0, 1, 2): {
(0, 1, 2, 9): 1
},
(0, 1, 2, 3): {
(0, 1, 2, 3, 4): 2
},
(0, 1, 2, 6): {
(0, 1, 2, 3): 1
},
(0, 1, 2, 9): {
(0, 1, 2, 6): 1
},
(0, 1, 7): {
(0, 1, 2): 2
},
(0, 1, 7, 9): {
(0, 1, 2, 9): 2
},
(0, 5): {
(0, 1): 1,
(0, 5, 7): 2
},
(0, 5, 7): {
(0, 5, 7, 9): 1
},
(0, 5, 7, 9): {
(0, 1, 7, 9): 1
},
(0, 8): {
(0, 5): 1
}
})
if (G.is_isomorphic(GT) != True):
return False
else:
print "G2 example passed."
# Some examples from Hong--Kang:
# type C, ex. 8.3.5, pg. 189
C = ClassicalCrystalOfAlcovePaths(['C', 3], [0, 0, 1])
G = C.digraph()
GT = DiGraph({
(): {
(0): 3
},
(0): {
(0, 6): 2
},
(0, 1): {
(0, 1, 3): 3,
(0, 1, 7): 1
},
(0, 1, 2): {
(0, 1, 2, 3): 3
},
(0, 1, 2, 3): {
(0, 1, 2, 3, 8): 2
},
(0, 1, 2, 3, 4): {
(0, 1, 2, 3, 4, 5): 3
},
(0, 1, 2, 3, 8): {
(0, 1, 2, 3, 4): 2
},
(0, 1, 3): {
(0, 1, 3, 7): 1
},
(0, 1, 3, 7): {
(0, 1, 2, 3): 1,
(0, 1, 3, 7, 8): 2
},
(0, 1, 3, 7, 8): {
(0, 1, 2, 3, 8): 1
},
(0, 1, 7): {
(0, 1, 2): 1,
(0, 1, 3, 7): 3
},
(0, 6): {
(0, 1): 2,
(0, 6, 7): 1
},
(0, 6, 7): {
(0, 1, 7): 2
}
})
if (G.is_isomorphic(GT) != True):
return False
else:
print "C3 example passed."
# type B, fig. 8.1 pg. 172
C = ClassicalCrystalOfAlcovePaths(['B', 3], [2, 0, 0])
G = C.digraph()
GT = DiGraph({
(): {
(6): 1
},
(0): {
(0, 7): 2
},
(0, 1): {
(0, 1, 11): 3
},
(0, 1, 2): {
(0, 1, 2, 9): 2
},
(0, 1, 2, 3): {
(0, 1, 2, 3, 10): 1
},
(0, 1, 2, 3, 10): {
(0, 1, 2, 3, 4): 1
},
(0, 1, 2, 9): {
(0, 1, 2, 3): 2,
(0, 1, 2, 9, 10): 1
},
(0, 1, 2, 9, 10): {
(0, 1, 2, 3, 10): 2
},
(0, 1, 5): {
(0, 1, 2): 3,
(0, 1, 5, 9): 2
},
(0, 1, 5, 9): {
(0, 1, 2, 9): 3,
(0, 1, 5, 9, 10): 1
},
(0, 1, 5, 9, 10): {
(0, 1, 2, 9, 10): 3
},
(0, 1, 8): {
(0, 1, 5): 3
},
(0, 1, 8, 9): {
(0, 1, 5, 9): 3,
(0, 1, 8, 9, 10): 1
},
(0, 1, 8, 9, 10): {
(0, 1, 5, 9, 10): 3
},
(0, 1, 11): {
(0, 1, 8): 3
},
(0, 7): {
(0, 1): 2,
(0, 7, 11): 3
},
(0, 7, 8): {
(0, 7, 8, 9): 2
},
(0, 7, 8, 9): {
(0, 1, 8, 9): 2
},
(0, 7, 8, 9, 10): {
(0, 1, 8, 9, 10): 2
},
(0, 7, 11): {
(0, 1, 11): 2,
(0, 7, 8): 3
},
(6): {
(0): 1,
(6, 7): 2
},
(6, 7): {
(0, 7): 1,
(6, 7, 11): 3
},
(6, 7, 8): {
(0, 7, 8): 1,
(6, 7, 8, 9): 2
},
(6, 7, 8, 9): {
(6, 7, 8, 9, 10): 1
},
(6, 7, 8, 9, 10): {
(0, 7, 8, 9, 10): 1
},
(6, 7, 11): {
(0, 7, 11): 1,
(6, 7, 8): 3
}
})
if (G.is_isomorphic(GT) != True):
return False
else:
print "B3 example 1 passed."
C = ClassicalCrystalOfAlcovePaths(['B', 3], [0, 1, 0])
G = C.digraph()
GT = DiGraph({
(): {
(0): 2
},
(0): {
(0, 1): 1,
(0, 7): 3
},
(0, 1): {
(0, 1, 7): 3
},
(0, 1, 2): {
(0, 1, 2, 8): 2
},
(0, 1, 2, 3): {
(0, 1, 2, 3, 5): 1,
(0, 1, 2, 3, 9): 3
},
(0, 1, 2, 3, 4): {
(0, 1, 2, 3, 4, 5): 1
},
(0, 1, 2, 3, 4, 5): {
(0, 1, 2, 3, 4, 5, 6): 2
},
(0, 1, 2, 3, 5): {
(0, 1, 2, 3, 5, 9): 3
},
(0, 1, 2, 3, 5, 9): {
(0, 1, 2, 3, 4, 5): 3
},
(0, 1, 2, 3, 9): {
(0, 1, 2, 3, 4): 3,
(0, 1, 2, 3, 5, 9): 1
},
(0, 1, 2, 5): {
(0, 1, 2, 3, 5): 2
},
(0, 1, 2, 8): {
(0, 1, 2, 3): 2
},
(0, 1, 2, 8, 9): {
(0, 1, 2, 3, 9): 2
},
(0, 1, 7): {
(0, 1, 2): 3,
(0, 1, 7, 8): 2
},
(0, 1, 7, 8): {
(0, 1, 7, 8, 9): 3
},
(0, 1, 7, 8, 9): {
(0, 1, 2, 8, 9): 3
},
(0, 2): {
(0, 1, 2): 1,
(0, 2, 5): 2
},
(0, 2, 5): {
(0, 2, 5, 8): 1
},
(0, 2, 5, 8): {
(0, 1, 2, 5): 1
},
(0, 7): {
(0, 1, 7): 1,
(0, 2): 3
}
})
if (G.is_isomorphic(GT) != True):
return False
else:
print "B3 example 2 passed."
# type B, fig. 8.3 pg. 174
return True
